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Basically assume that the entirety of the population of Holy Terra is evacuated so that no one will be endangered when the Emperor stops holding the Chaos forces back allowing him time to recover from his wounds. How will this go?

I think you are severely underestimating the potential size of the Warp Rift that could be created in such scenario.

According to Guilliman the Emperor can still speak to people in the throne room too. So if this was a viable plan he probably would've recommended it a long time ago.

As Tyran says, the warp rift would probably be way too big.

Regarding the evacuation:
It's crazy how long it takes to evacuate a Hive world. Imagine you could evacuate 1 mio people per hour (which would be incredible keeping in mind that shuttles/ships have to land, be boarded and fly away, including food, repair, refuelling for the ships), you could evacuate 24 million people per day, 8.76 billion per year. As terra has a population "in the hundreds of billions", this would mean between 11.4 and 114 years for an evacuation. Even if you could somehow raise it to 1 Mio per minute (how???) you would need between 70-700 days.
Meanwhile take in mind, that all those people would have to go somewhere. There are few systems that could handle 24 mio dudes (or 1.44 billion with the 1mio/minute plan) arriving within a day.

So in my personal opinion and judging from what I read about the empire so far: I assume the imperiums would see its recources "wiser" spend by only evacuating the 10-100 mio most important personell and write the rest of as martyrs.

It's the imperium. They seriously wouldn't bother. They'd get the important people out and tell the rest to fight for the emperor and dwell in His eternal glory.

Read "Dead men walking" for your answer.

Good god, just disseminating that message across Terra would be a miracle. There are probably slums that span continents. There are probably spaces that go as deep into the earth as far as they reach into the sky. Entire societies live and die on Terra without knowing anything of the outside world. I don't think Terra even has the infrastructure to achieve any form of population mobility.

I have to agree. Evacuating Terra would be impossible. It is probably the most populated planet in the Imperium filled with people whom the Imperium would not consider worth the resources to evacuate.

More importantly, the morale blow to the Imperium of even attempting to evacuate Holy Terra would destroy the Imperium. It is the beating heart of the Imperium. If Terra falls, the Imperium falls, and humanity falls.